Energy Minimization
I am doing undergraduate research in a lab that does molecular dynamics simulations and it struck me yesterday that the minimization of energy functions could be done with the methods I learned in vector calculus if one was to apply constraints to a system of equations with 3N variables, where N is the number of atoms.I went through a book on molecular modeling yesterday to see if it said anything regarding this and it stated that it is not generally possible to do that "for molecular systems due to the complicated way in which the energy varies with the coordinates." It then states that minimization is done with numerical methods, but it does not elaborate on why the way that the energy is modeled prevents use of the extreme value theorem.
Would someone elaborate on the nature of problems where the extreme value theorem cannot be used for minimization of a system of variables? If I do not get a response here, I will ask in the lab on Monday, but people here seem to be really good at applied mathematics, so I was hoping that someone here could elaborate on what the book said. 
[QUOTE=ShiningArcanine;131421]
Would someone elaborate on the nature of problems where the extreme value theorem cannot be used for minimization of a system of variables? [/QUOTE] My understanding is that the complexity of the minimization job depends on how accurately you describe the forces involved, and whether subatomic forces are included in that. Your book says that because the common wisdom holds that using gradientor Hessianbased minimization is impractical when there are tens of thousands of variables that all contribute to the minimum. Just calculating the forces that every atom exerts on all the others is a big job, and there is a large research literature on doing the computation efficiently. Finally, if you're looking for a global minimum somewhere inside a space that is very full of local minima, then you may never find it without massively sampling the entire space for the correct set of initial conditions needed. Certainly nobody expects a nonnumercial solution to such large problems, at least without simplifying away most of the detail that makes the problem relevant. But I could be wrong, so ask your lab anyway. 
Thank you so much for the information. I did not have much of a chance to go to my lab on Monday. Yesterday, I was there to get a book on Fortran. Today, I will be going there for a lab meeting, so I will ask then.

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